Three-term asymptotics for discrete Hardy--Rellich constants in high dimension

Abstract

Sharp Hardy--Rellich constants are spectral thresholds for operators with critical inverse-power potentials. Let $C_\ell(N)$ denote the optimal constant in the $\ell$th-order discrete Hardy--Rellich inequality on $\Z^N$. Recent independent work established the leading high-dimensional behavior $C_\ell(N)\sim2^\ell N^\ell$ for every fixed $\ell$. We determine the next two orders and prove \[ C_\ell(N)=2^\ell N^\ell+\gamma_\ell N^{\ell-1} +\delta_\ell N^{\ell-2}+O_\ell(N^{\ell-3}), \] with explicit coefficients $\gamma_\ell$ and $\delta_\ell$. The central difficulty is a degeneracy that grows with the dimension: after conjugation, the leading operator is scalar on the $2N$ nearest neighbours of the origin. We resolve this cluster using signed-permutation symmetry and an effective operator on four lattice orbits. Weighted torus estimates and Feshbach--Schur reduction justify the finite-dimensional expansion inside the full operator, while a residual bound and Temple's inequality give the stated remainder. In particular, $C_1(N)=2N-4-20/(3N)+O(N^{-2})$.